Process artichoke hearts in a food processor until finely chopped, set aside.
Melt 1 stick of butter in a large sauce pan over medium heat.
Process onion in a food processor until finely shredded.
Add shredded onion to the melted butter and sauté onion for 6-8 minutes until soft and translucent.
Add remaining ½ stick of butter salt, pepper, Greek seasoning, and garlic. Stir to combine.
Add chopped mushrooms.
Cook over medium heat until the mushrooms release their moisture and all liquid has cooked off, about 5-8 minutes.
Add the flour to the pan and stir to combine. Mixture will be very thick. Cook over medium heat for 2-3 minutes, stirring constantly.
Slowly add chicken broth, whisking to combine.
Pull pan off the heat and add the milk and heavy cream, whisking together until smooth.
Return pan to the heat.
Stir in processed artichoke hearts and sliced green onions. Reserve a few green onions for garnish.
Taste and season with additional salt if needed.
Cook soup over medium heat stirring frequently until soup thickens and is heated through. Do not boil soup, or it may separate.
To serve top each bowl with freshly cracked black pepper and sliced green onions.
Serve with rolls, biscuits or a crusty loaf of bread.